Definition: A function $f:\real^d \to \real$ is said to be continuous at a point $\p$ if for all $\eps > 0$, there exists $\delta > 0$:
\[\norm{\x-\p} < \delta \implies \abs{f(\x) - f(\p)} < \eps.\]A direct consequence of this definition is that the limit of any continuous function is simply the function evaluated at its limit point:
Theorem: A function $f:\real^d \to \real$ is continuous at a point $\x$ if and only if
\[\lim_{\x_n \to \x} f(\x_n) = f(\x).\]Indeed, the above theorem is sometimes taken to be the definition of a continuous function.